\subsection{}
Some thought about the kinetic energy and interaction energy.  The attractive interation energy, especailly this constant strength one, wants to spread the distribution of $\varphi_\vk$ evenly over $\vk$.  However, this encounter the problem from the kinetic energy, which always want to put the particle into the lowest energy state.  In the constant density of state, the kinetic energy increase is not as bad as it in the 3-D density of state, which has denser states in high energy. 